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Permutations (1 Viewer)
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Would this help?
Thank you
ExtremelyBoredUser
Bored Uni Student
Consider the thing in cases?
All cases that would be neglected:
1 - ND Together: 2 * 6! =1440
2 - 1 letter between them: 2! * 5! * 5 =1200
3 - 2 letter between them: 2 (N and D) * 5c2 (5 letters to choose from remaining) * 2 (2 ways to arrange them in the spaces between N and D) * 4! (Rearranging all groups) = 960
When you find all these cases, just subtract it from n!
7! - (1440+1200+960) = 1440
There might be a faster way but this is what I normally do for these type of questions,.
All cases that would be neglected:
1 - ND Together: 2 * 6! =1440
2 - 1 letter between them: 2! * 5! * 5 =1200
3 - 2 letter between them: 2 (N and D) * 5c2 (5 letters to choose from remaining) * 2 (2 ways to arrange them in the spaces between N and D) * 4! (Rearranging all groups) = 960
When you find all these cases, just subtract it from n!
7! - (1440+1200+960) = 1440
There might be a faster way but this is what I normally do for these type of questions,.
ExtremelyBoredUser
Bored Uni Student
Which question exactly?
ExtremelyBoredUser
Bored Uni Student
This is how you would construct the question:
9 _ _ _ _ _ _ _
A:
There's 7 slots and 10 choices for all of them. 10^7
B:
Last digit would only have odd numbers so only 5 choices which are 1, 3, 5, 7, 9 and hence 5C1 (choose 1 odd number) * 10^6 (Rest choices of spots)
C:
Odd digits only so therefore we only have 5 options 1, 3, 5, 7, 9. Therefore 5^7
D:
Odd even means First Odd Second Even Third Odd and so on. Thats the pattern The first number is already done for us (odd) so keep that in mind. There's 4 alternating spots for even and 3 alternating spots for even;
9 E O E O E O E
- E is even number
- O is odd number
For the even numbers, the options we have is 2,4,6,8 whereas for odd numbers we have 1,3,5,7,9. Therefore even has 4 choices, odd has 5 choices.
Therefore 4^4 * 5^3 = 32000
I did this a bit quick so I might've made a mistake, please check with answers or anyone else proof-check it.
EDIT: Typo fixed. Before it was 4^3, now it is correct 4^4
9 _ _ _ _ _ _ _
A:
There's 7 slots and 10 choices for all of them. 10^7
B:
Last digit would only have odd numbers so only 5 choices which are 1, 3, 5, 7, 9 and hence 5C1 (choose 1 odd number) * 10^6 (Rest choices of spots)
C:
Odd digits only so therefore we only have 5 options 1, 3, 5, 7, 9. Therefore 5^7
D:
Odd even means First Odd Second Even Third Odd and so on. Thats the pattern The first number is already done for us (odd) so keep that in mind. There's 4 alternating spots for even and 3 alternating spots for even;
9 E O E O E O E
- E is even number
- O is odd number
For the even numbers, the options we have is 2,4,6,8 whereas for odd numbers we have 1,3,5,7,9. Therefore even has 4 choices, odd has 5 choices.
Therefore 4^4 * 5^3 = 32000
I did this a bit quick so I might've made a mistake, please check with answers or anyone else proof-check it.
EDIT: Typo fixed. Before it was 4^3, now it is correct 4^4
Last edited:
In D this part '4^3 * 5^3 = 32000' when I put that into the calculator I got 8000, am I missing something?
It's meant to be 4^4 I'm pretty sure, as follows:
1, 2, 3, 4, 5, 6, 7, 8, 9
n in form 9 E O E O E O E = 4 x 5 x 4 x 5 x 4 x 5 x 4 = 32000 numbers
Last edited:
ExtremelyBoredUser
Bored Uni Student
Sorry its a typo. Its 4^4
There doesn't seem to be a Question 4 in your attachment.
For part (g):
The 3 yes's are always together, so we arrange them as one quantity (let's call it y). Note that the only possible ways of arranging (Y Y Y) in a line is 1.
N N N N N N N y
This is now a typical arrangement (making 'words'). Traditionally, the answer here is now 8!/(7!) = 8 ways.
The 3 yes's are always together, so we arrange them as one quantity (let's call it y). Note that the only possible ways of arranging (Y Y Y) in a line is 1.
N N N N N N N y
This is now a typical arrangement (making 'words'). Traditionally, the answer here is now 8!/(7!) = 8 ways.